316 A. Mukliopadhyay — Bijferential "Equation of all Parabolas. [No. 4, 



XII.— On the Bifferetifial Equation of all Parabolas. 

 By AsuTOSH MuKHOPADHYAT, M. A., F. R. A. S., F. R. S. E. 



[Received May 18th ;— Read June 6th, 1888.] 



Contents. 



§ 1. Introduction.* 



§ 2. Transon's Theory of Aberrancy. 



§ 3. Geometric Interpretation. 



§ 4. Miscellaneons Theorems. 



§ 1. Introduction, 

 It is my object in the present paper to give tlie geometrical inter- 

 pretation of the differential equation of all parabolas, as promised at 

 the end of my remarks on Monge's Differential Equation to all Conies. f 

 I have already incidentally pointed out]: the easiest method of deriving 

 the differential equation of all parabolas from the integral equation of 

 the curve, viz., the parabola being given by 



ax'^ + 2]ixy + by^ + 2gx + 2/^ + c = 0, 

 where Jfi =^ ab^ 



■we have, by solving for ?/, 



by=.-{Jix +/) ± {2 (hf-bg)x + (P-hc)}^^ 

 which may bo written 



y = Pcc -{- Q± \/rx + s , 



and this being on both sides operated upon by I -y) , leads to 

 whence 



(S)"'= 



lx + ', 

 so that 



\dx) \dxi) ""^' 



which is equivalent to the developed form 



d^y d'y _ /dhjY_. 



"^ ZT^ dx^ \d7^J -^' 



and this is the differential equation to be geometrically interpreted. It 



« For a full analysis of this paper, see P. A. S. B. 1888, pp. 156-157. 



t P. A. S. B. 1888, p. 86, footnote. 



X J. A. S. B. 1887, vol. Ivi, part ii, p. 136 j P. A. S. B. 1887, pp. 185-186. 



